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A histogram is an approximate representation of the
distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...

of numerical data. It was first introduced by
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...

. To construct a histogram, the first step is to " bin" (or "
bucket A bucket is typically a watertight, vertical cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism ( ...
") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent and are often (but not required to be) of equal size. If the bins are of equal size, a rectangle is erected over the bin with height proportional to the
frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...
—the number of cases in each bin. A histogram may also be
normalized Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
to display "relative" frequencies. It then shows the proportion of cases that fall into each of several
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

, with the sum of the heights equaling 1. However, bins need not be of equal width; in that case, the erected rectangle is defined to have its ''area'' proportional to the frequency of cases in the bin. The vertical axis is then not the frequency but ''frequency density''—the number of cases per unit of the variable on the horizontal axis. Examples of variable bin width are displayed on Census bureau data below. As the adjacent bins leave no gaps, the rectangles of a histogram touch each other to indicate that the original variable is continuous. Histograms give a rough sense of the density of the underlying distribution of the data, and often for
density estimation In probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a nu ...
: estimating the
probability density function and probability density function of a normal distribution . Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the ''x''-axis are all 1, then a histogram is identical to a relative frequency plot. A histogram can be thought of as a simplistic
kernel density estimation In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
, which uses a
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
to smooth frequencies over the bins. This yields a probability density function, which will in general more accurately reflect distribution of the underlying variable. The density estimate could be plotted as an alternative to the histogram, and is usually drawn as a curve rather than a set of boxes. Histograms are nevertheless preferred in applications, when their statistical properties need to be modeled. The correlated variation of a kernel density estimate is very difficult to describe mathematically, while it is simple for a histogram where each bin varies independently. An alternative to kernel density estimation is the average shifted histogram, which is fast to compute and gives a smooth curve estimate of the density without using kernels. The histogram is one of the seven basic tools of quality control. Histograms are sometimes confused with bar charts. A histogram is used for continuous data, where the bins represent ranges of data, while a
bar chart A bar chart or bar graph is a chart or graph that presents Categorical variable, categorical data with Rectangle, rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizo ...

is a plot of
categorical variable In statistics, a categorical variable is a variable (research), variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal catego ...
s. Some authors recommend that bar charts have gaps between the rectangles to clarify the distinction.

# Examples

This is the data for the histogram to the right, using 500 items: The words used to describe the patterns in a histogram are: "symmetric", "skewed left" or "right", "unimodal", "bimodal" or "multimodal". Symmetric-histogram.png, Symmetric, unimodal Skewed-right.png, Skewed-left.png, Bimodal-histogram.png, Bimodal Multimodal.png, Multimodal Symmetric2.png, Symmetric It is a good idea to plot the data using several different bin widths to learn more about it. Here is an example on tips given in a restaurant. Tips-histogram1.png, Tips using a $1 bin width, skewed right, unimodal Tips-histogram2.png, Tips using a 10c bin width, still skewed right, multimodal with modes at$ and 50c amounts, indicates rounding, also some outliers The
U.S. Census Bureau The United States Census Bureau (USCB), officially the Bureau of the Census, is a principal agency of the U.S. Federal Statistical System, responsible for producing data about the American people and economy. The Census Bureau is part of th ...
found that there were 124 million people who work outside of their homes. Using their data on the time occupied by travel to work, the table below shows the absolute number of people who responded with travel times "at least 30 but less than 35 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time. The problem of reporting values as somewhat arbitrarily is a common phenomenon when collecting data from people. : This histogram shows the number of cases per
unit interval In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
as the height of each block, so that the area of each block is equal to the number of people in the survey who fall into its category. The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers, with Q in thousands. : This histogram differs from the first only in the
vertical Vertical may refer to: * Vertical direction In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...
scale. The area of each block is the fraction of the total that each category represents, and the total area of all the bars is equal to 1 (the fraction meaning "all"). The curve displayed is a simple density estimate. This version shows proportions, and is also known as a unit area histogram. In other words, a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies: the height of each is the average frequency density for the interval. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also contiguous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)

# Mathematical definitions

The data used to construct a histogram are generated via a function ''m''''i'' that counts the number of observations that fall into each of the disjoint categories (known as ''bins''). Thus, if we let ''n'' be the total number of observations and ''k'' be the total number of bins, the histogram data ''m''''i'' meet the following conditions: : $n = \sum_^k.$

## Cumulative histogram

A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram ''M''''i'' of a histogram ''m''''j'' is defined as: : $M_i = \sum_^i.$

## Number of bins and width

There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is at least as old as Graunt's work in the 17th century, but no systematic guidelines were given until Sturges' work in 1926. Using wider bins where the density of the underlying data points is low reduces noise due to sampling randomness; using narrower bins where the density is high (so the signal drowns the noise) gives greater precision to the density estimation. Thus varying the bin-width within a histogram can be beneficial. Nonetheless, equal-width bins are widely used. Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb. The number of bins ''k'' can be assigned directly or can be calculated from a suggested bin width ''h'' as: :$k = \left \lceil \frac \right \rceil.$ The braces indicate the
ceiling function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
.

### Square-root choice

:$k = \lceil \sqrt \rceil \,$ which takes the square root of the number of data points in the sample (used by Excel's Analysis Toolpak histograms and many other) and rounds to the next
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.

### Sturges' formula

Sturges' formula is derived from a
binomial distribution In probability theory and statistics, the Binomial coefficient, binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, indep ...

and implicitly assumes an approximately normal distribution. :$k = \lceil \log_2 n \rceil+ 1 , \,$ Sturges' formula implicitly bases bin sizes on the range of the data, and can perform poorly if , because the number of bins will be small—less than seven—and unlikely to show trends in the data well. On the other extreme, Sturges' formula may overestimate bin width for very large datasets, resulting in oversmoothed histograms. It may also perform poorly if the data are not normally distributed. When compared to Scott's rule and the Terrell-Scott rule, two other widely accepted formulas for histogram bins, the output of Sturges' formula is closest when .

### Rice Rule

: The Rice Rule is presented as a simple alternative to Sturges' rule.

### Doane's formula

Doane's formulaDoane DP (1976) Aesthetic frequency classification. American Statistician, 30: 181–183 is a modification of Sturges' formula which attempts to improve its performance with non-normal data. :$k = 1 + \log_2\left( n \right) + \log_2 \left\left( 1 + \frac \right\right)$ where $g_1$ is the estimated 3rd-moment-
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of the distribution and :$\sigma_ = \sqrt$

### Scott's normal reference rule

Bin width $h$ is given by :$h = \frac,$ where $\hat \sigma$ is the sample
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v ...

. Scott's normal reference rule is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.

### Freedman–Diaconis' choice

The Freedman–Diaconis rule gives bin width $h$ as: :$h = 2\frac,$ which is based on the
interquartile range 250px, Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal Population In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The I ...
, denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.

### Minimizing cross-validation estimated squared error

This approach of minimizing integrated mean squared error from Scott's rule can be generalized beyond normal distributions, by using leave-one out cross validation: :$\underset \hat\left(h\right) = \underset \left\left( \frac - \frac \sum_k N_k^2 \right\right)$ Here, $N_k$ is the number of datapoints in the ''k''th bin, and choosing the value of ''h'' that minimizes ''J'' will minimize integrated mean squared error.

### Shimazaki and Shinomoto's choice

The choice is based on minimization of an estimated ''L''2
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:$\underset \frac$ where $\textstyle \bar$ and $\textstyle v$ are mean and biased variance of a histogram with bin-width $\textstyle h$, $\textstyle \bar=\frac \sum_^ m_i$ and $\textstyle v= \frac \sum_^ \left(m_i - \bar\right)^2$.

### Variable bin widths

Rather than choosing evenly spaced bins, for some applications it is preferable to vary the bin width. This avoids bins with low counts. A common case is to choose ''equiprobable bins'', where the number of samples in each bin is expected to be approximately equal. The bins may be chosen according to some known distribution or may be chosen based on the data so that each bin has $\approx n/k$ samples. When plotting the histogram, the ''frequency density'' is used for the dependent axis. While all bins have approximately equal area, the heights of the histogram approximate the density distribution. For equiprobable bins, the following rule for the number of bins is suggested: :$k = 2 n^$ This choice of bins is motivated by maximizing the power of a Pearson chi-squared test testing whether the bins do contain equal numbers of samples. More specifically, for a given confidence interval $\alpha$ it is recommended to choose between 1/2 and 1 times the following equation: :$k = 4 \left\left( \frac \right\right)^\frac$ Where $\Phi^$ is the
probit In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular Q–Q plot, exploratory statistical gra ...
function. Following this rule for $\alpha = 0.05$ would give between $1.88n^$ and $3.77n^$; the coefficient of 2 is chosen as an easy-to-remember value from this broad optimum.

### Remark

A good reason why the number of bins should be proportional to

# Applications

* In
hydrology Hydrology (from Ancient Greek, Greek wikt:ὕδωρ, ὕδωρ, ''hýdōr'' meaning "water" and wikt:λόγος, λόγος, ''lógos'' meaning "study") is the scientific study of the movement, distribution, and management of water on Earth and ...
the histogram and estimated
density function In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
of rainfall and river discharge data, analysed with a
probability distribution In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
, are used to gain insight in their behaviour and frequency of occurrence.An illustration of histograms and probability density functions
/ref> An example is shown in the blue figure. * In many
Digital image processing Digital image processing is the use of a digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people ...
programs there is an histogram tool, which show you the distribution of the
contrast Contrast may refer to: Science * Contrast (vision), the difference in color and light between parts of an image * Contrast (statistics), a combination of averages whose coefficients add up to zero, or the difference between two means * Behavioral ...
/ brightness of the
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s.

*
Data binning Data binning (also called Discrete binning or bucketing) is a data pre-processing technique used to reduce the effects of minor observation errors. The original data values which fall into a given small interval, a bin, are replaced by a value re ...
*
Density estimation In probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...
**
Kernel density estimation In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
, a smoother but more complex method of density estimation * Entropy estimation * Freedman–Diaconis rule *
Image histogram An image histogram is a type of histogram that acts as a graphical representation of the Lightness (color), tonal distribution in a digital image. It plots the number of pixels for each tonal value. By looking at the histogram for a specific imag ...
*
Pareto chart A Pareto chart is a type of chart that contains both bars and a line graph In the mathematical discipline of graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structur ...
*
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* V-optimal histograms

# References

* Lancaster, H.O. ''An Introduction to Medical Statistics.'' John Wiley and Sons. 1974.

Exploring Histograms
an essay by Aran Lunzer and Amelia McNamara

''(location of census document cited in example)''
Smooth histogram for signals and images from a few samples

* ttp://2000.jukuin.keio.ac.jp/shimazaki/res/histogram.html A Method for Selecting the Bin Size of a Histogram
Histograms: Theory and Practice
some great illustrations of some of the Bin Width concepts derived above.

Interactive histogram generator

Matlab function to plot nice histograms

Dynamic Histogram in MS Excel
* Histogra
construction
an
manipulation
using Java applets, an

on SOCR
Toolbox for constructing the best histograms
{{Statistics, descriptive Statistical charts and diagrams Quality control tools Estimation of densities Nonparametric statistics Statistics articles needing expert attention Frequency distribution